Minimal Conductivity in Bilayer Graphene

József Cserti

Eötvös University Department of Physics of Complex Systems

International School, MCRTN’06, Keszthely, Hungary, Aug. 27- Sept. 1, 2006.

J. Cs.: cond-mat/0608219

Near zeros concentrations the longitudinal conductivity is of the order of

Independent of temperature and magnetic field

Minimal Conductivity in Bilayer GrapheneK. S. Novoselov, E. McCann, S. V. Morozov, V. I. Fal'ko, M. I. Katsnelson, U. Zeitler, D. Jiang, F. Schedin, A. K. Geim, Nature Physics 2, 177 (2006)

Theoretical results for single layer graphene

Single layer graphene:

• A. W. W. Ludwig, M. P. A. Fisher, R. Shankar, and G. Grinstein, PRB 50, 7526 (1994) • E. Fradkin, PRB 63, 3263 (1986)• P. A. Lee, PRL 71, 1887 (1993)• E. V. Gorbar, V. P. Gusynin, V. A. Miransky, and I. A. Shovkovy, PRB 66, 045108 (2002) • V. P. Gusynin and S. G. Sharapov, PRL 95, 146801 (2005)• N. M. R. Peres, F. Guinea, and A. H. Castro Neto, PRB 73, 125411 (2006)• M. I. Katsnelson, Eur. J. Phys B 51, 157 (2006)• J. Tworzyd lo, B. Trauzettel, M. Titov, A. Rycerz, C.W.J. Beenakker, PRL 96, 246802 (2006)

K. Ziegler, cond-mat/0604537.

K. Nomura and A. H. MacDonald, cond-mat/0606589.

L. A. Falkovsky and A. A. Varlamov, cond-mat/0606800.

Short range scatteringCoulumb scattering

M. Koshino and T. Ando, cond-mat/0606166

M. I. Katsnelson, cond-mat/0606611

Theoretical results for bilayer graphene

strong-disorder regime

weak-disorder regime

E. McCann and V. I. Fal'ko, Phys. Rev. Lett. 96, 086805 (2006)

Hamiltonian for bilayer graphene

J=1 single layerJ=2 bilayer graphene

Equivalent form:

Pseudo spin, Pauli matrices

Plane wave solution:

Eigenvalues:

Green’s function:

Dirac cone

2 by 2 matrix

Kubo formula

conductivity tensor:

correlation function:

where

Fermi function:

A. Bernevig, PRB 71, 073201 (2005) (derived for spintronic systems)

Result

per valley per spin

where

Equivalent form:

A. W. W. Ludwig, M. P. A. Fisher, R. Shankar, and G. Grinstein, PRB 50, 7526 (1994)

Second method

Result

per valley per spin

Including the two valleys and the electron spin (factor of 4)

Kubo formula

Second method

The two definitions yield two different results for the longitudinal conductivity of perfect graphenes

But numerically they are close to each other

• The conductivity proportional with number of layers (J)

• Single layer graphene (J=1):

Our result using the 2nd method agrees with many earlier predictions

• Our result for bilayer is close to the experimental one

• Our result agrees with M. Koshino and T. Ando (cond-mat/0606166) result derived for the case of strong disorder • The two methods give two different results for the longitudinal conductivity !?!

• The minimal conductivity in graphene systems still remains a theoretical challenge in the future

Conclusions